Home
Class 12
MATHS
The range of f(x)=[1+sinx]+[2+s in2/x]+[...

The range of `f(x)=[1+sinx]+[2+s in2/x]+[3+s in x/3]++[n+s in x/n]AAx in [0,pi]` , where [.] denotes the greatest integer function, is, `{(n+n-2^2)/2,(n(n+1))/2}` `{(n(n+1))/2}` `{(n^2+n-2^)/2,(n(n+1))/2(n^2+n+2)/2}` `[(n(n+1))/2,(n^2+n+2)/2]`

Text Solution

Verified by Experts

`f(x)=[1+sinx]+[2+"sin"(x)/(2)]+[3+"sin"(x)/(2)]+...+[n+"sin"(x)/(n)] `
for, ` x in (0, pi)`
` sinx, "sin"(x)/(2),"sin"(x)/(3),"sin"(x)/(4),"sin"(x)/(5), … "sin"(x)/(n) in (0,1)`
` :. " for " x in (0,pi)`
`f(x)=1+2+3+ …+ n=(n(n+1))/(2)`
For `x=0, f(x)=1+2+3+...+n=(n(n+1))/(2)`
for ` x=pi, f(x)=1+[2+1}+3+4+...+n`
`=(1+2+3+...+n)+1`
`=(n(n+1))/(2)+1=(n^(2)+n+2)/(2)`
` :. " Range is " {(n(n+1))/(2),(n^(2)+n+2)/(2)}`
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise Solved Examples|15 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.1|15 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE ENGLISH|Exercise Archives (Numerical Value Type)|3 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE ENGLISH|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

The range of f(x)=[1+sinx]+[2+sin(x/2)]+[3+sin (x/3)]+....+[n+sin (x/n)]AAx in [0,pi] , where [.] denotes the greatest integer function, is,

show that [(sqrt(3) +1)^(2n)] + 1 is divisible by 2^(n+1) AA n in N , where [.] denote the greatest integer function .

The value o lim_(xtooo)(1/(n^(3)))([1^(2)x+1^(2)]+[2^(2)x+2^(2)]+…..+[n^(2)x+n^(2)]) is where [.] denotes the greatest integer function.

The value of int_0^x[cost]dt ,x in [(4n+1)pi/2,(4n+3)pi/2]a n dn in N , is equal to where [.] represents greatest integer function. pi/2(2n-1)-2x pi/2(2n-1)+x pi/2(2n+1)-x (d) pi/2(2n+1)+x

If a_(1) is the greatest value of f(x) , where f(x) =((1)/(2+[sinx])) (where [.] denotes greatest integer function) and a_(n-1)=((-1)^(n-2))/((n+1))+a_(n) (where n in N), then lim_(nrarroo) (a_(n)) is

Prove that for n=1, 2, 3... [(n+1)/2]+[(n+2)/4]+[(n+4)/8]+[(n+8)/16]+...=n where [x] represents Greatest Integer Function

The period of f(x)=[x]+[2x]+[3x]+[4x]+[n x]-(n(n+1))/2x , where n in N , is (where [dot] represents greatest integer function) then which of the following is correct a. n (b) 1 (c) 1/n (d) none of these

1+n. (2n)/(1+n)+(n(n+1))/(1.2) ((2n)/(1+n))^2+…..

lim_(n->oo)(1/(n^2+1)+2/(n^2+2)+3/(n^2+3)+....n/(n^2+n))

Evaluate lim_(ntooo) n^(-n^(2))[(n+2^(0))(n+2^(-1))(n+2^(-2))...(n+2^(-n+1))]^(n) .